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1、Problem IntroductionIs convergent or not?0nnaqIs convergent or not?11pnnDefinition of the positive term series Definition:1nnu(Positive term series)Necessary and sufficient conditions for convergence(monotonically increase)there are only two conditions:.nsssnn lim(1)when,n 1 diverge.nnu(2)has upper
2、bound,ns(positive constant)namelyns，0nu 12 nsss21ForFundamental Theorem()nssNote:The positive term series can be arbitrarily added brackets,its convergence(divergence)is invariable.If the positive term series converges,its sum does not change.The positive term series convergesThe partial sum of the
3、sequencensis bounded.Example 1Discuss the series converges or not.1121nn121 n1211211212 nnSn2121212 n211 By Theorem A,So,21n 1 the positive term series converges.Note:To determine a positive term series is convergent or not,we can compare it with another convergent(or divergent)positive term series.
4、Theorem B:Integral TestLet be a continuous,positive,non-increasing function on the interval 1,and suppose that=for all positive integers.the improper integral converges.1f x dx1kkaThen the infinite series converges if and only if If thenTheorem CComparison method,nnuv0 1nnvconverges1nnuconverges1nnu
5、diverges1nnvdiverges12and nnsuuuTheorem CProof:1Let nnvnnvu That is,the partial sum is bounded.1nnunvvv 21converges.If then,nnuv0 1.nnvconverges1nnuconvergesTheorem C,nnuvIfdivergesdivergesProof:then nnsLet()nsn and nnuv Not bounded 1nnvdiverges.,nnuv1nnv1nnu0If then,nnuv0 1nnudiverges1nnvdivergesDi
6、sadvantage of comparison method:There must be a reference series.1If nnu(diverges)convergesand()nnvku nN()nnkuv1then nnvconverges(diverges).DeductionExample 2convergesif 1.111111234ppppnn pppnns131211 211dd1nppnxxxx 11If 1,then ppnxnnx nnpnx1dIf 1,p pn1 nnpxx1d11npdxx )11(1111 pnp111 pnsso is,bounde
7、dnsthen seriespconverges.11npnExample 2diverges if 1.111111234ppppnn If 1,p then seriesp 11nndivergennp11 Use comparison method,diverge 11npnCommon comparison series(1)geometric seriesWe should know some convergent positive term series as comparison standards when we use the comparison method.(2)p-s
8、eries(3)harmonic series0when 1,convergewhen 1,divergennqaqqwhen 1,convergewhen 1,divergepp 11npn nnn13121111divergeSummary of the Positive SeriesDefinition of the positive term series Bounded Sum TestFundamental TheoremIntegral TestTheorem CCommon comparison seriesQuestions and AnswersShow that the series converges.1111!2!3!Aim:Show that the partial sum is bounded.!1 2 3nn 1 2 22 12n111!2nnThus,11111!2!3!nSn111111242n12 12n2By the Bounded Sum Test,the given series converges.Positive Series:The Integral Test